# Tagged Questions

Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, Fourier transform and the discrete Fourier transform (and relatives). The non-commutative analog is (representation-theory).

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### intuition behind an identity related to fourier transforms

I saw the proof of this identity in a question about Fourier transforms : $F(f(−t))w=F(f(t))(−w)$ Can someone give the intuition behind it ? What I understand of Fourier transform of a function ...
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### Fourier series problems

I've got an "interesting" problem. I've gotten a way through it, but I'd like someone to look if what I've done so far is correct, and what to do next. We've got a function that is $0$ on the ...
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### Sobolev spaces and Holder continuity (or, fractional derivatives and singularities)

I have two specific questions. The first is the result I actually need, and the second would let me prove it. EDIT: The second statement was wrong. I am keeping it for posterity. I am adding a third ...
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### How to get the real frequency from an FFT output graph?

I am working in Origin 7.0 and trying to understand how the FFT Analysis works. In an attempt to understand, I decided to run a test of the program using known frequencies. The function I used was: 2*...
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### Fourier transform of function similar to a Riesz kernel

I am trying to prove that the Fourier transform of $$\frac{x_1 x_2} {|x|^4}$$ in $\mathcal{R}^2$ (in the sense of distributions) is a bounded multiplier given by $\frac{\xi_1 \xi_2}{|\xi|^2}$ but am ...
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### Eigenfunctions of the Laplace-Beltrami operator of a torus

The eigenfunctions of the Laplace-Beltrami operator of the flat torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$ and their multiplicity are well-known. What happens if we change the sides of the torus ...
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### Uniform bound on Fourier series

This is from Fourier Analysis by Stein and Shakarchi, section 3, exercise 19. I am trying to prove that $\sum_{0<|n|\le N} e^{inx}/n$ is uniformly bounded in $N$ and $x\in [-\pi,\pi]$. Following ...
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### convolution of three functions of two variables

Give three functions of two variables $a(x,y),b(x,y),c(x,y)$ one can construct the following convolution like integral: $y(x,y) = \int dx' dy' a(x',y')b(x-x',y') c(x-x',y-y')$ which I have a hard ...
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### Fourier Transform, and identities

I need to calculate the fourier transform of this $t \cdot sin(t-3)+2 \cdot t \cdot cos(3t) \cdot rect(6t)$ is the following valid, based on the first fourier identity i've read in a book, and ...
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### Showing that complex exponentials of the Fourier Series are an orthonormal basis

I am revisiting the Fourier transform and I found great lecture notes by Professor Osgood from Standford (pdf ~30MB). On page 30 and 31 he show that the complex exponentials form an orthonormal ...
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Solve this via Fourier method: $$u_t-u_{xx}=0 \quad\quad 0< x<\pi, \quad t >0,$$ $$u(0,t)=u_x(\pi,t)=0, \quad\quad t \ge 0$$ $$u(x,0)=2\sin\left(\frac{3x}{2}\right) \, \cos{2x},... 1answer 55 views ### What is Fourier Space I know a some basics stuff regarding Fourier Analysis (Fourier series and Fourier transforms), but I've seen the term "Fourier Space" come up and I'm having trouble finding a definition for what this ... 0answers 99 views ### an “alternate derivation” of Poisson summation formula and discrete Fourier transformation Inspired by this post, I am trying to do a derivation of a Poisson summation formula. My starting point is this:$$ \frac{1}{2\pi} \int^{\infty}_{-\infty} e^{i k x} dx=\delta(k) $$I simply wish ... 1answer 112 views ### A naive example of discrete Fourier transformation We know a discrete Fourier transformation with discrete n and continuous x_1,x_2:$$ \sum_{n\in\mathbb{Z}} e^{-in(x_1-x_2)\frac{2\pi}{L}}=L\delta(x_1-x_2) $$with Dirac delta function \delta. ... 1answer 21 views ### How to find the Fourier components I've got a sinusodial signal: $$\Delta I=\cos(\Delta \omega t + \varphi)).$$ and I would like to rewrite it as the sum of a cosine and a sine signal(without a phaseterm): ... 1answer 99 views ### Show that the Fourier transform of a a distribution is C^{\infty} I am trying to understand the solution to the following problem: Let u \in \mathcal{D}'(\mathbb{R}^{n}) such that u(x) = c \log(|x|) when |x|>1, where c \in \mathbb{C}. Show that u \in \... 1answer 57 views ### Fourier and Z transform of a signal? We have$$X(k)=4[u(k-2)-u(k)* d(k-3)] I need to find the Fourier transform,$Z$ transform,as well as dhe magnitude and phase spectra. First of all I think that I need to convert the $u(k)$ and $u(k-... 1answer 142 views ### Convolution of ring Delta function Assume$f(r)=\delta(r-R)$where$\delta(\cdot)$is a ring delta function. In other word,$f$is a circular delta function on a circle with radius$R$. I want to do the convolution of$f$with itself ($...
I'm trying to get to grips with the relationship between a signal $x(t)$, its Fourier transform $X(F)$ and the graph representation of $t$ plotted against $|X(F)|$. I've been using Matlab to perform ...